NCERT Solutions Ganita Prakash (Part 1) Chapter 4 –1054.5 Playing with Quadrilaterals — Joining Triangles

Book page 104 Updated on2026-09-05

Q1.

Take two cardboard cutouts of an equilateral triangle of sidelength 8 cm. Can you join them to get a quadrilateral?

Answer

Yes. Place the two cutouts so that one side of the first lies exactly on one side of the second, with the triangles on opposite sides of that common edge.

Every side is 8 cm, so any side of one triangle fits any side of the other. The common edge disappears into the interior and becomes a diagonal, and the four outer edges — 8 cm each — become the sides of a quadrilateral.

2 triangles × 3 sides = 6 edges
2 edges are used up as the join ⇒ 6 – 2 = 4 sides ✓

Why they must be joined along a whole edge: if you slide one triangle so that only part of an edge touches, the outline gains extra corners and stops being four-sided. The edges must match end to end, which is why equal side lengths matter.

Q2.

What type of a quadrilateral is this? Justify your answer.

Answer

It is a rhombus with side 8 cm and angles 60°, 120°, 60°, 120°.

All four outer edges are sides of equilateral triangles ⇒ all equal 8 cm ⇒ rhombus

Angles: at the two ends of the common edge, two 60° angles meet:
60° + 60° = 120°
At the other two vertices the angle is a single triangle angle: 60°
Check: 60 + 120 + 60 + 120 = 360° ✓

It is not a square, since 60° ≠ 90°.

Why it cannot be anything else: four equal sides is precisely the definition of a rhombus. The diagonal that used to be the join is 8 cm — equal to a side — so it makes two equilateral triangles, which is what fixes the angles at 60° and 120°.

Q3.

Take two cardboard cutouts of an isosceles triangle with sidelengths 8 cm, 8 cm, and 6 cm. What are the different ways they can be joined to get a quadrilateral?

Answer

A join is possible along any pair of edges of the same length, and for each such pair the second triangle may be flipped over or turned round. That gives three genuinely different quadrilaterals.

Join along	How the second piece is placed	Resulting sides	Quadrilateral
the 6 cm edge	mirror image (or half turn — both give the same figure here)	8, 8, 8, 8	rhombus
an 8 cm edge	half turn about the midpoint of the edge	8, 6, 8, 6	parallelogram
an 8 cm edge	mirror image across the edge	8, 6, 6, 8	kite

The book shows the first two of these.

Why the mirror and the half turn agree on the 6 cm edge: the triangle is isosceles with the 6 cm side as its base, so it is already symmetric about the perpendicular bisector of that base. Flipping it makes no difference to the outline. On an 8 cm edge there is no such symmetry, so the two placements give different figures.

Q4.

What quadrilaterals are these? Justify your answers.

Answer

First figure — joined along the 6 cm edge: a rhombus of side 8 cm.

All four outer edges are the 8 cm sides ⇒ four equal sides ⇒ rhombus
The 6 cm join becomes a diagonal, and it bisects the two angles it meets
(each half-triangle is isosceles, so the diagonal is a line of symmetry)

Second figure — joined along an 8 cm edge, one triangle given a half turn: a parallelogram with sides 8 cm and 6 cm.

Top and bottom edges: 6 cm each Left and right edges: 8 cm each
Opposite sides equal ⇒ parallelogram (Question 9 of the next Figure it Out proves this implication)

More directly: a half turn about the midpoint of the shared 8 cm edge sends each side to a parallel copy pointing the opposite way, so opposite sides are parallel. That is the definition of a parallelogram.

Why a half turn always gives a parallelogram: rotating a triangle by 180° about the midpoint of one side reverses every direction. The image of the 6 cm side therefore runs parallel to the original 6 cm side, and the image of the third side runs parallel to that third side — both pairs of opposite sides come out parallel.

Q5.

Take two cardboard cutouts of a scalene triangle with sides 6 cm, 9 cm, and 12 cm. What are the different ways they can be joined to get a quadrilateral?

Answer

Since all three sides are different, a join is only possible along matching edges — 6 with 6, 9 with 9, or 12 with 12. For each of the three edges there are two placements (a half turn or a mirror image), so there are six ways in all.

Join along	Half turn gives sides	Mirror gives sides
6 cm	9, 12, 9, 12	9, 12, 12, 9
9 cm	6, 12, 6, 12	6, 12, 12, 6
12 cm	6, 9, 6, 9	6, 9, 9, 6

Tip: the mirror joins may need care. Reflecting across the longest edge can push the reflected vertex so that the outline caves in — you then get a concave quadrilateral rather than a convex one. Cut the pieces out and try each of the six before deciding.

Q6.

Are you able to identify the different quadrilaterals that are obtained by joining the triangles? Justify your answer whenever you identify a quadrilateral.

Answer

Two clear families appear.

The three half-turn joins give parallelograms. A half turn about the midpoint of the shared edge reverses all directions, so each remaining side becomes parallel to its partner:

join along 6 cm ⇒ parallelogram with sides 9 cm and 12 cm
join along 9 cm ⇒ parallelogram with sides 6 cm and 12 cm
join along 12 cm ⇒ parallelogram with sides 6 cm and 9 cm

The three mirror joins give kites (when the outline stays convex), because reflection produces two adjacent pairs of equal sides:

join along 12 cm ⇒ sides 6, 9, 9, 6 ⇒ a kite with diagonal 12 cm

This last one is the figure the book uses on page 105 to introduce the kite. Its adjacent sides are equal in pairs — 6 cm with 6 cm at the top, 9 cm with 9 cm at the bottom.

Why reflection gives a kite and rotation gives a parallelogram: a reflection keeps the shared edge as a line of symmetry, so the two sides on either side of it are mirror partners — adjacent equal pairs, which is a kite. A half turn instead sends each side to the opposite corner, so the equal pairs end up opposite each other — which is a parallelogram.

Check it yourself: whichever way you join them, the four angles must total 360°, because the two triangles contribute 180° each.

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